Fastest-convergence conjecture for the complementing flip process

Let $\mathcal{R}$ be a convergent flip process of order $k$ with a single destination. For $\delta\in(0,\frac{1}{2})$, let $\tau_{\mathcal{R}}^-(\delta)$ denote its lower convergence time.

Fastest-convergence conjecture. For every such $\mathcal{R}$ and every $\delta\in(0,\frac{1}{2})$,

$$\tau_{\mathcal{R}}^-(\delta)\ge \frac{-\ln(2\delta)}{2(k)_2}.$$

The bound is attained by the complementing flip process, whose convergence time is described in the source as half that of an ignorant process with output density $1/2$. The source does not state whether the conjecture has been resolved.